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Talk Outline How did we find this? What is it good for? What else have we done? spacetime approach pulsars, general asymptotic behavior of “free” magnetospheres? push the spacetime approach as far as possible… Field sheet picture No ingrown hair theorem varia In this talk:

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The Michel Monopole Michel 1973 In split form, can regard as a)toy model for dipole pulsar b)model for asymptotic pulsar magnetosphere from Beskin 2010 or

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An Observation The Michel monopole has charge and current satisfying In spacetime language, the charge-current four-vector is lightlike or null. This observation is powerful. Null structure is fundamental to spacetime.

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Null-Current Solutions We assumed null current and used the Newman-Penrose formalism, which is designed to leverage null structure. (Brennan, SEG, Jacobson 2013) We found the general solution with null, radial current in flat, Schwarzschild, and Kerr (radial=PNC) spacetimes. We later re-derived the solutions using the language of differential forms, where the calculations are simple and the physical content is clear (to the forms-initiated!) (SEG & Jacobson 2014) forms example:

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New Solutions The general outgoing solution with radial, null current in flat spacetime is non-stationary & non-axisymmetric! (Also valid in Schwarzschild, and we found analogous solutions in Kerr) magnetic monopoleoutgoing Poynting flux Michel monopole: 3+1:

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2. Asymptotic Magnetosphere Conjecture: If outgoing boundary conditions are imposed at infinity, then for any interior boundary conditions the current will be asymptotically null and radial. Related to Michel’s “minimal torque condition” Numerical evidence from Kalapotharakos et al 2011 Evidence against from Tchekhovskoy et al 2013? If so, these exact solutions describe the generic far-field behavior of force-free magnetospheres. Could model outgoing kink in field lines originating from a glitch or catastrophic event. (any other evidence for/against?)

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3. Non-Scattering Waves Ingoing version in Schwarzschild (Eddington-Finklestein coordinates) The free function can specify an initial wave packet… …and the wave just goes in. This was shocking to us! All other kinds of radiation scatter off the black hole., In particular there are no vacuum electromagnetic waves like this.

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Field sheet metric Particles* move on timelike geodesics of the field sheets Alfven wave group velocity follows field sheet null geodesics The spacetime metric induces a metric on each field sheet. In stationary axisymmetry, every field sheet has a Killing vector, Thus there is a conserved quantity along field sheet geodesics. This is the quantity used to calculate particle wind (e.g. CKF) Where k is null is called a light surface. It’s a causal boundary for particle and Alfven propagation. (SEG&Jacobson 2014) field line angular velocity *neglecting cyclotron, drift, and curvature radiation reaction

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Extremal Znajek Condition In the stream equation formulation for stationary, axisymmetric force-free fields in Kerr, there is a regularity condition on the horizon (Znajek 1977), This condition is not sufficient for regularity in the extremal case a=M. There we have an additional condition (SEG&Jacobson2014),

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Stream Equation for No Poloidal Field The usual Grad-Shafranov approach is built on the assumption of a non- vanishing poloidal magnetic field. If the field vanishes you have a different stream equation (SEG&Jacobson 2014) Chi is the electrostatic potential, See related work of Contopoulos (1995)